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(Abridged:) It has been hypothesized that the location of Herbig Ae/Be stars (HAeBes) within the empirical relation between the inner disk radius (r$_{in}$), inferred from K-band interferometry, and the stellar luminosity (L$_*$), is related to the presence of the innermost gas, the disk-to-star accretion mechanism, the dust disk properties inferred from the spectral energy distributions (SEDs), or a combination of these effects. This work aims to test whether the previously proposed hypotheses do, in fact, serve as a general explanation for the distribution of HAeBes in the size-luminosity diagram. GRAVITY/VLTI spectro-interferometric observations at $\sim $2.2 $ \mu$m have been obtained for five HBes representing two extreme cases concerning the presence of innermost gas and accretion modes. V590 Mon, PDS 281, and HD 94509 show no excess in the near-ultraviolet, Balmer region of the spectra ($\Delta$D$_B$), indicative of a negligible amount of inner gas and disk-to-star accretion, whereas DG Cir and HD 141926 show such strong $\Delta$D$_B$ values that cannot be reproduced from magnetospheric accretion, but probably come from the alternative boundary layer mechanism. Additional data for these and all HAeBes resolved through K-band interferometry have been compiled from the literature and updated using Gaia EDR3 distances, almost doubling previous samples used to analyze the size-luminosity relation. We find no general trend linking the presence of gas inside the dust destruction radius or the accretion mechanism with the location of HAeBes in the size-luminosity diagram. Underlying trends are present and must be taken into account when interpreting the size-luminosity correlation. Still, it is argued that the size-luminosity correlation is most likely to be physically relevant in spite of the previous statistical warning concerning dependencies on distance., Comment: 20 pages, 17 figures. The main data reduction and analysis process ia available at https://github.com/marcosarenal/gravity-data-reduction

(Abridged) Herbig Ae/Be stars (HAeBes) have so far been studied based on relatively small samples that are scattered throughout the sky. Their fundamental stellar and circumstellar parameters and statistical properties were derived with heterogeneous approaches before Gaia. Our main goal is to contribute to the study of HAeBes from the largest sample of such sources to date, for which stellar and circumstellar properties have been determined homogeneously from the analysis of the spectral energy distributions (SEDs) and Gaia EDR3 parallaxes and photometry. Multiwavelength photometry was compiled for 209 bona fide HAeBes for which Gaia EDR3 distances were estimated. Using the Virtual Observatory SED Analyser (VOSA), photospheric models were fit to the optical SEDs to derive stellar parameters, and the excesses at infrared (IR) and longer wavelengths were characterized to derive several circumstellar properties. The stellar temperature, luminosity, radius, mass, and age were derived for each star based on optical photometry. In addition, their IR SEDs were classified according to two different schemes, and their mass accretion rates, disk masses, and the sizes of the inner dust holes were also estimated uniformly. The initial mass function fits the stellar mass distribution of the sample within 2 < M_star/M_sun < 12. In this aspect, the sample is therefore representative of the HAeBe regime and can be used for statistical purposes when it is taken into account that the boundaries are not well probed. A statistical analysis was carried out mainly focused on the different circumstellar properties of Herbig Ae (HAe) and Be (HBe) stars. The complete dataset is available online through a Virtual Observatory-compliant archive, representing the most recent reference for statistical studies on the HAeBe regime., 45 pages, 13 figures, 3 tables. Accepted in A&A